Rational algebraic expression
An expression of the form $\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x)\neq 0$.
Examples:
$$
\frac{x}{x+3}, \quad \frac{2 x^2-3 x+1}{x-4}, \quad \frac{5}{x^2}, \quad \frac{x^2-9}{x^2+2 x+1}
$$
Non-examples:
We treat these like numerical fractions, but we must also pay attention to values that make the denominator zero.
A rational expression is undefined when its denominator is zero.
Restriction
A restriction is any value of the variable that is not allowed because it makes the denominator zero.
$$
\frac{x+5}{x-2}
$$
$$
x \neq 2
$$
$$
\frac{3 x}{x(x-1)}
$$
Always state restrictions before simplifying or cancelling any factors, because simplification can hide the values that make the original denominator zero.
To simplify a rational expression:
Simplify $\frac{x^2-9}{x^2+4 x+3}$.
Solution
These are the easiest operations: treat them like numerical fractions.
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